The last problem on which Curry worked before he died in 1982 was that
of defining a reduction in combinatory logic to correspond closely to the
usual beta-reduction in lambda-calculus. He did not succeed. Several solutions
to this problem have been posed since then, but despite some ingenuity
in their formulation, none has been really clean and simple enough to make
its development attractive. I believe the task of finding a workable combinatory
beta-reduction is one of the main unsolved problems in combinatory logic.
(It is not a "tidy" problem and it promises no beautiful solution -- but
then, neither does real life!) In this talk I shall discuss criteria
for acceptability of a beta reduction, and describe the known candidates,
and suggest how far they succeed or fail in satisfying these.